From asymptotic distribution and vague convergence to uniform convergence, with numerical applications

Let ${\Lambdan={\lambda{1,n},\ldots,\lambda{dn,n}}}n$ be a sequence of finite multisets of real numbers such that $dn\to\infty$ as $n\to\infty$, and let $f:\Omega\subset\mathbb R^d\to\mathbb R$ be a Lebesgue measurable function defined on a domain $\Omega$ with $0<\mud(\Omega)<\infty$, where $\mud$ is the Lebesgue measure in $\mathbb R^d$. We say that ${\Lambdan}n$ has an asymptotic distribution described by $f$, and we write ${\Lambdan}n\sim f$, if [ \lim{n\to\infty}\frac1{dn}\sum{i=1}^{dn}F(\lambda{i,n})=\frac1{\mud(\Omega)}\int\Omega F(f({\boldsymbol x})){\rm d}{\boldsymbol x}\qquad\qquad(*) ] for every continuous function $F$ with bounded support. If $\Lambdan$ is the spectrum of a matrix $An$, we say that ${An}n$ has an asymptotic spectral distribution described by $f$ and we write ${An}n\sim\lambda f$. In the case where $d=1$, $\Omega$~is a bounded interval, $\Lambdan\subseteq f(\Omega)$ for all $n$, and $f$ satisfies suitable conditions, Bogoya, B\"ottcher, Grudsky, and Maximenko proved that the asymptotic distribution (*) implies the uniform convergence to $0$ of the difference between the properly sorted vector $[\lambda{1,n},\ldots,\lambda{dn,n}]$ and the vector of samples $[f(x{1,n}),\ldots,f(x{dn,n})]$, i.e., [ \lim{n\to\infty}\,\max{i=1,\ldots,dn}|f(x{i,n})-\lambda{\taun(i),n}|=0, \qquad\qquad(**) ] where $x{1,n},\ldots,x{dn,n}$ is a uniform grid in $\Omega$ and $\taun$ is the sorting permutation. We extend this result to the case where $d\ge1$ and $\Omega$ is a Peano--Jordan measurable set (i.e., a bounded set with $\mud(\partial\Omega)=0$). See the rest of the abstract in the manuscript.